Find the absolute maximum and absolute minimum values of the function f(x)=(x-1)((x-5)^3)+11 (A) Interval = [1,4].Absolute maximum = Absolute minimum = (B) Interval = [1,8].Absolute maximum =...
Find the absolute maximum and absolute minimum values of the function
f(x)=(x-1)((x-5)^3)+11
(A) Interval = [1,4].
Absolute maximum =
Absolute minimum =
(B) Interval = [1,8].
Absolute maximum =
Absolute minimum =
(C) Interval = [4,9].
Absolute maximum =
Absolute minimum =
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
You need to find the critical values of functon, hence, you need to solve the equation f'(x) = 0 such that:
`f'(x) = (x-5)^3 + 3(x-1)(x-5)^2`
You need to solve the equation f'(x) = 0 such that:
`(x-5)^3 + 3(x-1)(x-5)^2 = 0`
You need to factor out `(x-5)^2 ` such that:
`(x-5)^2(x - 5 + 3x - 3) =` 0
`(x-5)^2 = 0 =gt x_(1,2) = 5`
`4x - 8 = 0 =gt 4x = 8 =gt x_3 = 2`
a) You need to select the end values of interval [1,4] to evaluate if derivative is positive or negative at these values such that:
`x = 1 =gt f'(1) = (1-5)^2(4-8) = 16*(-4) = -64lt0`
`x=4 =gt f'(4) = (4-5)^2(16-8) = 1*8 =8gt0`
Hence, the function reaches its absolute minimum over interval [1,4] at x = 2.
b) Notice from a) that the function decreases over [1,2] and it increases over [2,4], hence, you need to select the a value from interval [4,8] to evaluate if derivative is positive or negative such that:
`x =6 =gt f'(6) = (6-5)^2(24-8) = 1*(16) = 16gt0`
`x = 4.5 =gt f'(4.5) = (4.5-5)^2(18-8) gt 0`
Hence, the function reaches its absolute minimum at x = 2 and it has no absolute maximum over [1,8].
c) Notice that the function has no absolute maximum or minimum over [4,9].
Related Questions
- `f(x) = x + 1/x, [0.2, 4]` Find the absolute maximum and minimum values of f on the given...
- 1 Educator Answer
- `f(x) = (x^2 - 1)^3, [-1, 2]` Find the absolute maximum and minimum values of f on the...
- 1 Educator Answer
- `f(x) = 12 + 4x - x^2, [0, 5]` Find the absolute maximum and minimum values of f on the...
- 1 Educator Answer
- Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x+(9/x)...
- 1 Educator Answer
- `f(x) = x e^(-x^2/8), [-1, 4]` Find the absolute maximum and minimum values of f on the...
- 2 Educator Answers